From a0343985c103af90216907c6eba515615afccd43 Mon Sep 17 00:00:00 2001 From: arc Date: Fri, 5 Sep 2025 12:59:16 -0600 Subject: [PATCH] vault backup: 2025-09-05 12:59:16 --- .../Integration with Trig Identities.md | 13 +++++++++++-- 1 file changed, 11 insertions(+), 2 deletions(-) diff --git a/education/math/MATH1220 (calc II)/Integration with Trig Identities.md b/education/math/MATH1220 (calc II)/Integration with Trig Identities.md index 000a48b..6bbe4e7 100644 --- a/education/math/MATH1220 (calc II)/Integration with Trig Identities.md +++ b/education/math/MATH1220 (calc II)/Integration with Trig Identities.md @@ -22,7 +22,7 @@ $$ -\int(1 - 2u^2 + u^4)du $$ 6. Substituting $\cos(x)$ back in for $u$, we get the evaluated (but not entirely simplified) integral: $$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$ # Trigonometric Substitutions -Trigonometric substitution is useful for equations containing $\sqrt{a^2 + x^2}$ or $a^2 + x^2$, where $a$ is any constant. +Trigonometric substitution is useful for equations containing $\sqrt{a^2 + x^2}$ or $a^2 + x^2$, where $a$ is any constant. It removes any addition or subtraction. The general process involves the use of a trig identity and multiplying everything in that identity by a constant. @@ -41,4 +41,13 @@ This enables us to make use of **substitution** to simplify many integrals. $$ \int \frac{3}{4+x^2}dx = 3\int \frac{1}{4 + x^2}dx$$ 2. Let $x = 2tan\theta$ and $dx = (2sec^2\theta d\theta)$, substitute accordingly $$ = 3\int\frac{1}{4 + 4\tan^2\theta}(2\sec^2\theta)d\theta$$ -3. Factor $4$ \ No newline at end of file +3. Factor $4$ out in the denominator +$$ = 3\int\frac{1}{4(1 + \tan^2\theta)}(2\sec^2\theta)d\theta$$ +4. Considering the identity $1 + \tan^2 \theta = \sec^2 \theta$ +$$ = 3\int\frac{1}{4(\sec^2\theta)}(2\sec^2\theta)d\theta$$ +5. $\sec^2\theta$ is present in the numerator and the denominator, so we can cancel those out. This means that: +$$ 3\int\frac{2}{4}d\theta = \frac{3}{2} \theta + C$$ +6. At this point, we want to determine what $\theta$ is equal to relative to $x$. + 1. Look back to step 2 we defined $x = 2\tan\theta$ + 2. Moving $2$ to the other side to, +$$ $$ \ No newline at end of file